\subsection{Charge-dipole interaction: linear-switch form}
The switched potential energy is:
\begin{equation}
u^{\mathrm{S}}(r)=u(r)\cdot s(r)
\end{equation}
The modulating linear function $s$ is:
\begin{displaymath}
s(r)= 
\left\{ 
\begin{array}{lll}
1 &\mathrm{ if } \qquad r < r_{s}\\
(r_{c}-r)/(r_{c}-r_{s}) &\mathrm{ if }  \qquad r_{s} \leqslant r \leqslant r_{c}\label{eq:ssdmodu} \\
0 &\mathrm{ if } \qquad r > r_{c}\\
\end{array}
 \right. 
\end{displaymath}
The switched force (for $ r_{s} \leqslant r \leqslant r_{c}$) is: 
\begin{equation}
%\mathbf{f}=\frac{C(2r_c-r)}{r^3(r_c-r_s)}\hat{\mathbf{e}}= \frac{C}{r^3}\left[\left(\frac{3r_\mathrm{c}-2r}{r_\mathrm{c}-r_\mathrm{s}}\right)\frac{\mathbf{r}}{r}\cos\theta-\left(\frac{r_\mathrm{c}-r}{r_\mathrm{c}-r_\mathrm{s}}\right)\hat{\mathbf{e}}\right]
%\boxed{\mathbf{f}= \frac{C}{r^3}\left[\left(\frac{3r_\mathrm{c}-2r}{r_\mathrm{c}-r_\mathrm{s}}\right)\frac{\mathbf{r}}{r}\cos\theta-\left(\frac{r_\mathrm{c}-r}{r_\mathrm{c}-r_\mathrm{s}}\right)\hat{\mathbf{e}}\right]}
\mathbf{f}= \frac{C}{r^3(r_\mathrm{c}-r_\mathrm{s})}\left[\left(\frac{3r_\mathrm{c}}{r} - 2\right)\cos\theta\,\mathbf{r}+(r-r_\mathrm{c})\,\hat{\mathbf{e}}\right]
\end{equation}
The switched torque (for $ r_{s} \leqslant r \leqslant r_{c}$) is:
\begin{equation}
\mathbf{T}=\frac{C(r_c-r)}{r^3(r_c-r_s)}\mathbf{r}\times\hat{\mathbf{e}}
\end{equation}

